# Quotients of the Gordian and H(2)-Gordian graphs

@inproceedings{Flippen2021QuotientsOT, title={Quotients of the Gordian and H(2)-Gordian graphs}, author={C. Flippen and Allison H. Moore and Essak Seddiq}, year={2021} }

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the… Expand

#### References

SHOWING 1-10 OF 53 REFERENCES

Symmetric quotients of knot groups and a filtration of the Gordian graph

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2019

Abstract We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a… Expand

THE Ck-GORDIAN COMPLEX OF KNOTS

- Mathematics
- 2006

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3 by using "a crossing change". In this paper, we define the… Expand

Knot Graphs and Gromov Hyperbolicity

- Mathematics
- 2019

We define three types of knot graphs, constructed with the help of unknotting operations, the concordance relation and knot invariants. Some of these graphs have been previously studied in the… Expand

Jones polynomials and classical conjectures in knot theory

- Mathematics
- 1987

The primeness is necessary in the last statement ofTheorem B, since the connected sum of two figure eight knots is alternating, but it has a minimal non-alternating projection. Note that the figure… Expand

3-coloring and other elementary invariants of knots

- Mathematics
- 1998

Classical knot theory studies the position of a circle (knot) or of several circles (link) in R or S = R3∪∞. The fundamental problem of classical knot theory is the classification of links (including… Expand

A note on the Gordian complexes of some local moves on knots

- Mathematics
- 2018

In this paper, the ♯¯-move is defined. We show that for any knot K0, there exists an infinite family of knots {K0,K1,…} such that the Gordian distance d(Ki,Kj) = 1 and pass-move-Gordian distance… Expand

A polynomial invariant for knots via von Neumann algebras

- Mathematics
- 1985

Thus, the trivial link with n components is represented by the pair (l ,n), and the unknot is represented by (si$2 * * • s n i , n) for any n, where si, $2, • • • > sn_i are the usual generators for… Expand

THE GORDIAN COMPLEX OF KNOTS

- Mathematics
- 2002

In this paper, we define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an… Expand

Gromov hyperbolicity and a variation of the Gordian complex

- Mathematics
- 2009

We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus… Expand

State Models and the Jones Polynomial

- Mathematics
- 1987

IN THIS PAPER I construct a state model for the (original) Jones polynomial [5]. (In [6] a state model was constructed for the Conway polynomial.) As we shall see, this model for the Jones polynomial… Expand